Question

Let V and W be irreducible representations of $S_n$ and $S_m$ over a field of characteristic 0. Then the Littlewood-Richardson coefficients allow us to compute the isomorphism type of the induced $S_{n+m}$-module V⊗W↑. This induction comes from the inclusion

$S_n\times S_m \rightarrow S_{n+m}$.

Now suppose V=W. Then V⊗V↑ is a $S_{2n}$-module. But actually there's a symmetry coming from the symmetric monoidal category structure, so there is another induction up to an $S_{2n}$-module structure:

Extend the action of $S_n\times S_n$ on V⊗V by including the symmetry $c_V$, this naturally extends the group to the wreath product $S_n\sim S_2$. Induction along

$S_n\sim S_2 \rightarrow S_{2n}$

gives the representation that I want:

$(V\otimes V)_{S_n\sim S_2}\uparrow^{S_{2n}} \hookrightarrow V\otimes V\uparrow^{S_{2n}}$.

Using the Littlewood-Richardson rules we know the structure of the last term in terms of semi-standard skew tableau. My question is, how do we characterise the inclusion?

Answer 1

You want to read Splitting the square of a Schur function into its symmetric and antisymmetric parts, where this question is answered in terms of domino tableaux. I am also a big fan of Domino tableaux, Schützenberger involution, and the symmetric group action which, to my mind, gives the "right" formulation of the domino tableaux rule.

Answer 2

My understanding was that induction from wreath products was supposed to correspond to plethystic substitution of symmetric functions. But I don't really understand that, so I can't really say.

Answer 3

Is there a connection to zonal polynomials, or is that what David just said?